The Macro Model of the Inequality Process Explains a Quirky Aspect of U.S. Wage Incomes, 1961-2003: The Surge in Wage Income Nouveaux Riches
John Angle
Inequality Process Institute, Post Office Box 429, Cabin John, Maryland, USA

Abstract: The macro model of the Inequality Process (IP) is derived from the solution of the IP's micro model and approximates its stationary distribution. The micro model is a stochastic interacting particle system in which a positive quantity is exchanged between pairs of particles. The macro model is a gamma pdf whose parameters are given in terms of the IP's parameters. The IP version discussed in this paper is:

Randomly pair particles. Let one of these pairs be particle i and particle j. A fair coin is tossed and called. If particle i wins, it receives an ωθ share of particle j's wealth. If particle j wins, it receives an ωξ share of particle i's wealth. The other particle encounters are analogous. Repeat for all particles.

Particle i's ωξ's does not change. Whenever it loses, it loses an ωξ share of its wealth. The IP's metatheory relates ωξ's to worker productivity (more productivity, smaller ωξ). Angle (1983, 1993, 2003, and 2006) has shown that the IP explains a large number of quantitative and qualitative aspects of the distribution of wage income conditioned on productivity. In particular, Angle (1999, 2006) shows that the macro model of the Inequality Process provides a parsimonious fit to the U.S. wage income distribution conditioned on education. Such a model should account for all time-series of scalar statistics of annual wage income conditioned on education and for the time-series of all unconditional scalar statistics of wage income too. The present paper shows that the macro model of the Inequality Process (IP) does that, choosing as an example the quirky time-series that has roiled U.S. social science, alarmed the U.S. popular press, and provided rhetorical grist for one 2004 U.S. presidential candidate. The example is the rapid increase in wage income nouveaux riches in the U.S.


First-passage time probability distributions in interday FOREX
Kunal Bhattacharya
Satyendra Nath Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata-700098, India

Abstract: First-passage time probability distributions in interday FOREX. We have analyzed generalized first-passage time probability distributions in the interday foreign-exchange data for different countries ranging over many years. The fluctuations are assumed to be independent and shown to obey an exponential distribution. This allows to model a time series' as a random walk problem. It is seen that the first passage probability densities in the data can be explained reasonably well in the random walk regime. The above facts seem to true for almost all the countries. We conclude that even in the presence of separate background economies the random walk description for the first passage properties of the interday foreign exchange data is highly robust.


Generalized statistical models of economy markets
Marco Patriarca1, Els Heinsalu1,Anirban Chakraborti2
1Institute of Theoretical Physics, Tartu University, Tahe 4, 51010 Tartu, Estonia
2Department of Physics, Banaras Hindu University, Varanasi 221005, India

Abstract: We compare some versions of a statistical model, in which N units interact with each other exchanging a quantity x, according to a given microscopic random law depending on some saving parameters. This model can represent e.g. gas molecules colliding with each other and exchanging energy or economic agents carrying out transactions and exchanging wealth. A model of Angle is compared with that of Chakraborti and Chakrabarti and from numerical fitting an equilibrium gamma-distribution of orders n and n' respectively is suggested, with n = n'/2, for which the explicit dependence on the saving parameter is provided. We reformulate and study the various models in a way suitable to further generalizations.


Monetary wealth distributions under different modes of debt constraints
Siyan Chen, Keqiang Li, Yougui Wang
Department of Systems Science, School of Management, Beijing Normal University, Beijing, 100875, People's Republic of China

Abstract: In this paper the distribution of monetary wealth is studied based on a random transfer model by computer simulations and theoretical analysis. Debt is introduced to the model with different modes of constraints. With a limit to debt of each agent, it is shown that the stationary monetary wealth follows the exponential Boltzmann-Gibbs law. When the limit is set to the total volume of debt, the stationary distribution becomes an asymmetric Laplace one. In order to understand the basic mechanism that underlies the formation of these particular distributions, we apply the methods of statistical physics to deduce the exact expressions of the distributions, which are found to be in a good agreement with simulation results under different limitations over debt.


Multiphase transition of stock prices and stock index
Kishore Chandra Dash
Dept. of Physics, Neelashaila Mahavidyalaya, Rourkela 769042, Orissa, India

Abstract: It is very difficult to predict the movement of stock prices and indices because socio-economic systems are much more complex than physical systems even with fractal nature. Socio-economic systems are sensitive to different social, economic, natural or manmade disasters, positive or negative news, which can come unnoticed unlike that of physical systems. However it is observed that movement of stock prices rise or fall up to a certain level and remain almost stagnant with a small variation. The movement of stock prices is not observed just like the stagnancy of temperature when phase transition occurs in case of matters. In this paper I have tried to compare the price and index movement with that of the phase transition of matters when they are subjected to heat supply. However in case of stock prices and indices, there are multiple phase transitions. In this paper an attempt has been made to compare stock prices with that of temperature, demand with supply of heat, supply with extraction of heat, ratio of volume of transaction to the change in price with the specific heat and volume transacted during phase transition with latent heat. Transition due to evaporation and ebullition with the movement of stock prices has been applied. The comparison between the liquid and illiquid stock has been attempted. Although the chart of a stock with price vs. time looks like a gaussian, it contain a large number of approximated gaussians with fractals from which the future move of the stock price may be predicted. Prediction of stock market crash or boom from the chart has been discussed.


Debt-credit economic networks of banks and firms: the Italian case
Giulia De Masi
Economics Department, Polytechnic University of Marche, Piazza Martelli, 8 - 60121 Ancona, Italy

Abstract: The economic stability is related to the resilience of the whole system of banks and firms to single firms/banks failures. In general a single failure of a firm can trigger other events of failures of other firms. Moreover firms have loans with banks; therefore firms and banks are interconnected by complex relationships of debt-credit. This dependence exposes also the banks to a shortage of liquidity or the risk of failure in the worst cases, as a consequence of firms failures.

A relationship of credit-debt is present also between different banks through the inter-bank market. European banks provide the quantity of liquidity they need in weekly auctions with the European Central Bank; then banks can borrow and lend money each other in the inter-bank market, in order to fulfil unexpected withdrawals of customers and the requirement to maintain a certain deposit in the National Central Bank. In this work, the focus is on the one hand on the credit-debt dependences between Italian banks (inter-bank market), on the other hand on credit-debt dependences between Italian banks and firms. The Italian inter-bank market is an example of a transparent market where all agents (Italian and foreign banks) have exact information about transactions of other partners (time, volume, rate, even identity).

Even if a priori each agent can establish loans with any other, we observe from real data that transactions happen between partners with specific characteristics. We can represent this system using graph theory. We can naturally define a network where the nodes are banks and the links are the transactions between pairs of banks. A similar approach is used to represent the relationships of Italian firms with banks. A typical pattern is observed. Implications of the observed relationships architecture on the robustness of the whole system is analyzed.


Weighted Network for Scientific Collaboration of Econophysicists: Statistics and Evolution
Zengru Di
Department of Systems Science, School of Management, Beijing Normal University

Abstract: The development of Econophysics is studied from the perspective of scientific communication networks. Papers in Econophysics published recently are collected. Then a weighted and directed network of scientific communication, including collaboration, citation and personal discussion, is constructed. Its static geometrical properties, including degree distribution, weight distribution, weight per degree, betweenness centrality and community structure, give a nice overall description of the research works. Inspired by this empirical analysis of econophysicists network, an evolutionary model for weighted networks is proposed. Besides a new vertex added in at every time step, old vertices can also attempt to build up new links, or to reconnect the existing links. The number of connections repeated between two nodes is converted into the weight of the link. This provides a natural way for the evolution of link weight. The path-dependent preferential attachment mechanism with local information is also introduced. It increases the clustering coefficient of the network significantly. The model shows the scale-free phenomena in degree and vertex weight distribution. It also gives well qualitatively consistent behavior with the empirical results.


Chain Reaction in Market Process
Bijay Bal and Kuntal Ghosh
Saha Institute of Nuclear Physics, India

Abstract: We are aware of chain reaction in nuclear fission and Chemical reaction. But when a new product tries to enter into the market, it also passes through some similar steps of chain reaction. Chain reaction has the steps like its initiation process, its control and sustaining mechanism, management of waste etc. It may be common experience of daily passengers in train that at the onset it is very difficult to sale a single copy of a new item even after long introduction about the item by the salesman. But as soon as one or two copies of the item are sold in the compartment, a number of copies of the item are sold in no time. These phenomena may be assigned as a case of psychology driven initiation of chain reaction in market process . If the affinity for purchase is very near the critical barrier, the acts of nearest neighbors purchasing the item influences psychologically to overcome the barrier and purchase process follows chain reaction .The introduction of pouch in packaging consumable goods and a variation of its size ( for example) is one of the ways, how the height of the psychological barriers of the consumers can substantially be tuned. The sustenance of this chain reaction depends upon (in addition to the height of psychological barrier) the nearest neighbor separation and availability of money with individual at that occasion.

Typical character of sustenance of sale phenomena has critical and sub critical Natures. In critical range, the product itself is capable of creating enough appreciation among the consumers and which in turn acts as a positive feedback delighting the product for future sale and a self sustained sale mechanism is observed. In the sub critical region the product may not have the enough appreciation about it?s quality, but the product goes on sale mostly due to continuous attractive advertisement and subsidiary items attached with the product. This sub critical chain reaction persists in market so long as the adequate advertisement and other prize items are present with the product.

Waste management is another important aspect of market process. The excess production rate compared to the sale generates waste. A long term accumulation of this waste (if proper care is not taken in time) generates a negative feedback in market dynamics. As a result the overall cost of the product increases with time and ultimately the over burdened product may be slowly losing the entire market.


Relaxation Oscillation in the Character of Some Commodities in the Market
Bijay Bal and Kuntal Ghosh
Saha Institute of Nuclear Physics, India

Abstract: It is our common experience that a section of commodity (specially man designed food items like Chocolate, Rasgulla, Samosa and other consumable products like soap, perfumes, shampoo etc.) is found to have changed with time their size or volume keeping their geometrical shape and price per unit quanta of the item unchanged. With a particular price setting starting from a maximum size the item is found to be decreasing in size with time in a quasistatic way and regularly sold in market. But as the item thus reaches a critical size (~50% of maximum), it some way psychologically makes the purchaser unhappy (faces a psychological barrier) and the items come back in the market again with its earlier maximum size with price of unit quanta doubled. Again with time, the size goes on decreasing slowly keeping price per unit quanta of the item fixed to new scale and in this way the size of the item follows a typical pattern of relaxation oscillation with time. It should also be noted that under the guise of this process of relaxation oscillation the producer maintains a continuous increase of the price of unit mass of the ingredients of the product (approximately linearly with time) without any well thought significant opposition from the purchaser side by resorting to a puzzle between the quantum price and price of unit mass.


Weighted networks at Polish market
M. Chmiel, J. Sienkiewicz, K. Suchecki and J. A. Holyst
Faculty of Physics and Center of Excellence for Complex Systems Research Warsaw University of Technology, Koszykowa 75, PL 00-662 Warsaw, Poland

Abstract: In the present study we consider relations between companies in Poland taking into account common branches they belong to. Using a commercial database we construct a bipartite graph of companies and branches. It is clear that companies belonging to the same branch compete for similar customers, so the market induces interactions between them. Link weights describe the number of common branches for a companies pair. On the other hand two branches can be related by companies acting in both of them. To remove weak, accidental links we use a concept of "threshold filtering" for weighted networks where link weights correspond to a number of existing connections (common companies or branches) between a pair of nodes. Using cutoffs of links weights we construct networks with different filtering levels and study degree distributions of such networks. We also investigate Gibbs entropy of degree distributions to find out the amount of information we filter out.


Autocatalytic networks and economic growth: A mathematical model
Sanjay Jain
Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India



Multifractal Properties of the Indian Stock Market
Sunil Kumar1,2 and Nivedita Deo1
1Department of Physics and Astrophysics, University of Delhi, Delhi-110007
2Ramjas College, University of Delhi, Delhi-110007

Abstract: We report on a study of the Indian price index (BSE & NSE) from August, 2002 to February, 2007 using the Multifractal Detrended Fluctuation Analysis (MF-DFA). Numerically we find that the MF-DFA fluctuation function Fq(s) shows that the indices exhibit multifractal properties. The exponent h(q) related to the Hurst exponent H (h(2) is the Hurst exponent), changes with the moment q. The Hurst exponent is 0.47571 & 0.46855 for the BSE and NSE index respectively. In particular the dependence of the multifractal scaling exponent τ(q) as a function of q has been explicitly found numerically which shows multifractality. We also observe that h(q) of these non-stationary time series are sensitive to the presence of a crash or a boom.


Social Networks: Examples of Deterministic and Stochastic Processes Modeling
Chitro Majumdar1 and René Algesheimer2
1ARW Business House -Brussels, Louizalaan 479 - Box 50, Avenue Louise, B-1050 Brussels, Belgium
2University of Zurich, Switzerland

Abstract: Nowadays there is much activity on (social) networks, mainly the work of economists, mathematicians and statistical physicists. In social networking the modelings that are deterministic and stochastic. I have applied deterministic techniques to small-world networks and scale-free networks (social networks can be either of these), but the uses of deterministic modeling are limited. The two main characteristics of small-world networks are strong local clustering and small diameter.

Social networks are highly volatile. There has been a lot of recent research using stochastic models for these networks. We will give an example with the variables of a small data set of a virtual social network, where we are interested in economic variables for the company. The variables are not normally distributed, but power law distributed. Thus, as a basic modeling I guess the regression does not require normality. The critical values of the regression output, however, are only asymptotical. We could even use Quantile regression in order to reach some interesting results. We also could suggest some theoretical examples of stochastic processes (such as pure jump processes, branching processes, etc.) and taking into account the dynamic properties of social networks since the beginning.


Dynamic Financial Analysis as the untrodden path under Solvency-II
Chitro Majumdar
Institute of Mathematical Statistics and Actuarial Science-Bern, Switzerland

Abstract: Dynamic Financial Analysis (DFA) is the most advance modeling process in today's property and Casualty industry-allowing us to develop financial forecasts that integrate the variability and interrelationships of critical factors affecting our results.

DFA in the capital budgeting decision process of a company launching a new invention and predicting the impact of the strategic decision on the balance sheet in a horizon of few years. To recognize the few factors that will affect the asset liability cash flow are demand uncertainty, sales volatility, credit risk, volatility in the price of raw materials cost of capital to name a few. Each of these random variables can be stochastically simulated either based on the distribution of retrospective data or under strategic assumptions. When simulated in a combined way the future cash flows can be predicted which in return would dictate the capital requirements in the future. Depending on the capital structure of the company and simulated interest rate in the capital market, business cycles and the final earnings volatility of the company can be predicted to identify the return and associated risks where DFA would be risk-based insurance pricing, for insurance companies (under Solvency-II).


The International Trade Network: weighted network analysis and modelling
K. Bhattacharya1, G. Mukherjee1,2, J. Saramäki3, K. Kaski3, S. S. Manna1,3
1Satyendra Nath Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata-700098, India
2Bidhan Chandra College, Asansol 713304, Dt. Burdwan, West Bengal, India
3Laboratory of Computational Engineering, Helsinki University of Technology, P.O. Box 9203, FIN-02015 TKK, Finland

Abstract: The International Trade Network is studied from the weighted network perspective using the volumes of mutual annual trade between two countries as the edge weights. The analysis spans over 53 years and it shows that the weight distribution is well approximated by a log-normal distribution. Moreover, the strength of a node, which is the total trade volume of a country in a year, is observed to grow algebraically with the Gross Domestic Product over all years. In addition, a small number of richest countries is seen to form a strong community -- the ``rich-club'' -- whose internal trade comprises a major fraction of the total volume of international trade. Interestingly the size of the rich club over the time span of 53 years from 1948 to 2000 was found to shrink to less than half of its starting value. Many of the empirical features of ITN are reproduced by a simple model, the starting point of which is the well-known gravity model of international trade.


Econophysics: Calculus vs. Economic Standard Models
Juergen Mimkes
Physics Department, University of Paderborn, Germany

Abstract: Econophysics applies methods of physics to economics. "Ex post" and "ex ante" calculations and economic growth may be explained by "path dependent" integrals: Y - C = 0: At zero growth income (Y) and consumption costs (C) are equal. The total differential forms dY and dC are not path depending and are known "ex ante": the resulting output is zero. A Cobb Douglas function, Y = AKαL1-α may be assumed depending on capital K, labor L and technology A. Income and consumption per laborer, y = Y/L = Akα, will both grow according to dy/y = αdk/k + dA/A. Standard economic models may be valid for rich countries with (nearly) no economic growth.

Y - C ≠ 0: In economies with real growth income δY is a not exact differential form. Y and C are path dependent integrals and cannot be calculated "ex ante", without knowledge of the production process. A general Cobb Douglas function Y does not exist, standard economic models will not apply. Economic growth must be calculated from calculus: δy/k = αdk/k - df. The production function (- df) replaces technology (A) and corresponds to negative entropy, to reduction of disorder, to a better infrastructure of the economic system. These results will apply to poor countries with high increase of GDP.

The production function f introduces entropy to economics: Industry is a Carnot machine: Production of goods is entropy reduction (Δf < 0) at low price, trade is distribution of commodities or entropy production (Δf > 0) at high price. The Carnot process may act like a heat pump and draw capital from poor countries to rich countries, or run like a motor. The fuel for cars and economies is the same: oil.


Analytics and Processes of derivatives pricing
Jyotishka Dutta, Avishek Ghosh, Rohit Patel, Arnab Barat
Indian Statistical Institute, 203, B.T. Road, kolkata, 700108

Abstract: The finacial market around the world is a complex process. Now the more hot topic is the options market. The dynamics of this finantial quotes are noteable. Much work has been done to model them and try for prediction. We firstly try the knowledge of the stochastical processes and try to show them an simplified eto process and then, modify it to further complicated models. Thereafter we try pure and simple data analysis to suggest a time series model. Finally it's good to try and see volatility of different quotes and try to predict some futuristic trend. To explore,Understand and implement the methodology of polynomial tree used to price Multiasset derivatives.


Variations in Financial Time Series: Modelling through Wavelets and Genetic Programming
Dilip P. Ahalpara1, Prasanta K. Panigrahi2, Jitendra C. Parikh2
1Institute for Plasma Research, Near Indira Bridge, Gandhinagar-382428, India
2Physical Research Laboratory, Navrangpura, Ahmedabad-380009, India

Abstract: We analyze the variations in S & P CNX NSE daily closing index stock values through discrete wavelets. Transients and random high frequency components are effectively isolated from the time series. Subsequently, small scale variations as captured by Daubechies level 3 and 4 wavelet coefficients and modelled by genetic programming. We have smoothened the variations using Spline interpolation method, after which it is found that genetic programming captures the dynamical variations quite well through Pade type of map equations. The low-pass coefficients representing the smooth part of the data has also been modelled. We further study the nature of the temporal variations in the returns.


Empirical studies and models of income distributions in society
Peter Richmond
School of Physics, Trinity College, Dublin 2, Ireland

Abstract: We review early and more recent studies of income and wealth distributions in society. Various approaches to theoretical models will be assessed and compared in detail to data for the UK during the period 1992-2002.


Estimation of delay in information flow among stocks
M. S. Santhanam
Theoretical Physics and Complex Systems Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India

Abstract: We study BSE stock market data. It is known that there are groups of stocks that tend to display same kind of dynamics. From the studies on stock cross-correlations and random matrix theory, it is also known that there are individual stocks that are anti-correlated. We focus on these individual stocks and study their dynamics. They are (anti-) correlated even though they do not organically belong to the same industry. We estimate the delay in interactions between these individual stocks using cross-correlation and synchronization techniques. We propose a mechanism using driven coupled map lattices that can lead to such interactions.


Knowledge Sharing and R & D Investment
Abhirup Sarkar
Economics Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata - 700 035

Abstract: We consider an R & D race between two symmetric firms. The game consists of two stages. In the first stage, firms non-cooperatively decide upon their levels of investment in R & D which, in turn, determine the Poisson probabilities of their subsequent sucesses. In the second stage, they engage in a Nash bargaining game to share their knowledge. We show that the firms over-invest and earn lower profits if knowledge sharing is possible compared to the situation where it is not. Hence, before the first stage, if the firms are given the option of precommitting to no knowledge sharing, they will do so and be better off.


Recurrence analysis of financial time series
Apu Sarkar
Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, India

Abstract: Recurrence plot (RP) is a quite easy tool used in time-series analysis, in particular for measuring unstable periodic orbits embedded in a chaotic dynamical system. Recurrence quantification analysis (RQA) is an advanced tool that allows the study of intrinsic complexity of a dynamical system with a set of few parameters. These methods yield a deeper understanding of the underlying process of a given time series and are applicable to study nonstationary time series. In this work. we have used RP and RQA to study the dynamical behaviour of the Indian financial market. We have studied time series data from Indian foreign exchange market and Bombay stock market using these methodologies.


Dynamical structure of behavioral similarities of the market participants in the foreign exchange market
Aki-Hiro Sato
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University

Abstract: The tick frequencies for currency pairs in the foreign exchange market, defined by the number of tick quotation per unit time, are investigated. In order to measure the behavioral similarity of the market participants several measures (cross-correlation coefficients, phase correlation functions defined by similarity between two phase spectra, spectral distances defined by the Kullback Leibler divergence between two power spectra, and similarity between two spontaneous phases) are employed. From numerical simulations of the agent-based model of a financial market it is confirmed that these measures quantify similarities among agent parameters. By using these measures dynamical structure of behavioral similarities of the market participants in the foreign exchange market is calculated. These results imply that the market participants behave changing their parameters slowly. This finding is applicable to measuring risks of portfolio with high resolution.


Collective Behavior in the Indian Stock Market: Cross-correlation structure of stock movement in NSE
Sitabhra Sinha and Raj Kumar Pan
The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113

Abstract: In the first ever detailed study of the internal structure of a market other than NYSE, we analyze the interactions between the most frequently traded stocks in the National Stock Exchange (NSE) of India. We have performed an eigen-analysis of the cross-correlation structure using daily returns of 201 frequently traded stocks over a period of more than a decade, coinciding with the period of rapid transformation of the Indian market as it emerged to become one of the world's largest. We find absence of distinct sector identity in the market, when compared to NYSE, and the movement of stocks is dominantly influenced by the overall market trend. This observation fits with the general belief that emerging markets tend to be more correlated than developed markets. We also present a two-factor model of return dynamics, which shows how the intermediate eigenvalues depend systematically on the relative strength of sector identity, to substantiate that the Indian financial market does not have the internal structure of multiple groups of co-moving stocks as seen in developed markets like NYSE.


Is Inequality Inevitable in Organized Societies ?: Income distribution as a consequence of resource flow in hierarchical structures
Sitabhra Sinha
The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113

Abstract: Almost all organized societies, once they attain a certain level of complexity, exhibit inequality in terms of income of its members. Hierarchical stratification of social classes is the major contributor to such unequal distribution of income, with intra-class variation often being negligible compared to inter-class differences. In this talk, we will explore examples from different historical periods, ranging from medieval Byzantium to the Mughal empire of India in 17th century, and different kinds of organizations, from criminal gangs in USA to Manufacturing & IT Services companies in India, to show that hierarchical structure of the organization lies behind the observed income inequality in societies. As income distribution can therefore be seen as a consequence of resource flow in a hierarchically structured network, we will then present a model of how such organizations can arise in a system of interacting agents, and show that empirically observed income inequality can be explained as a consequence of the division of the assets inflow at various levels of such an organization.


Networks of firms and a ridge in the production space
Wataru Souma
NiCT/ATR CIS Applied Network Science Lab., 2-2-2, Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0288, Japan

Abstract: We consider complex networks in economics. As is well known, networks are constructed from nodes and links. In this study, we consider firms as nodes. Thus, links represent many kinds of relationships between firms. As these relationships, we consider shareholding, transaction, interlocking directors, and joint application of patent. Through network analysis, we clarify topological characteristics of these networks. In addition, we consider relationships between dynamics of firms and these networks, and propose mathematical models that can explain both of dynamics of firms and growth of networks.

We also consider dynamics of firms in the production space that is characterized by capital stock, employment, and profit. Hence, in this space, each firm moves to maximize profit by controlling of capital stock and employment. We show that dynamics of rational firms are described by a ridge equation. We analytically solve this equation by assuming the Cobb-Douglas production function, and obtain a solution. We discuss the efficiency of management of firms by comparing of this solution and empirical values.